ME 315 Exam 3 8:00-9:00 PM Thursday, April 16, 2009 CIRCLE YOUR DIVISION

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1 ME 315 Exam 3 8:00-9:00 PM Thurday, Aril 16, 009 Thi i a cloed-book, cloed-note examination. There i a formula heet at the back. You mut turn off all communication device before tarting thi exam, and leave them off for the entire exam. Pleae write legibly and how all work for your own benefit. Pleae how your final anwer in the boxe rovided. State all aumtion. Pleae arrange all your heet in the correct order. Make ure they are all included. Name: Lat Firt CIRCLE YOUR DIVISION Div. 1 (9:30 am) Prof. Murthy Div. (1:30 m) Prof. Choi Problem 1 (30 Point) (30 Point) 3 (40 Point) Total (100 Point) Score

2 1. (30 oint) Conider a heat exchanger with a heat exchange area of 30 m and oerating under the condition lited in the table below. Hot Fluid Cold Fluid Heat Caacity C (kw/k) 5 Inlet temerature o C Outlet temerature o C (i) Determine the outlet temerature of the hot tream in o C. T ho = (ii) I the heat exchanger oerating in a. Parallel flow b. Counterflow c. Can t tell Circle only one anwer and exlain your choice. Exlanation: (iii) Calculate the overall heat tranfer coefficient U (W/m K). U = (iv) What would the effectivene be if the heat exchanger were infinitely long? =

3 Solution (i) q m c T m c ( T T ) m c ( T T ) h, h h, i h, o c, c c, o c, i C ( T T ) C ( T T ) h h, i h, o c c, o c, i q ( kw / K)(57.5C 45 C) (5 kw / K)(60 C T ) 5kW ho, T h,o 55 C (ii) T c,o 57.5C T h,o 55C T c,o T h,o Counterflow (iii) q U AT lm C h (T h,i T h,o ) C c (T c,o T c,i ) 5 kw T lm T 1 T ln (T 1 T ) T 1 T h,i T c,o 60C 57.5C.5C T T h,o T c,i 55C 45C 10C T lm T 1 T.5C 10C ln (T 1 T ) ln (.5C 10C) C q U AT lm U (30 m )( C) 5 kw U kw m K W m K 3

4 (iv) q q max q max C min (T h,i T h,i ) (kw /K)(60C 45C) 30kW A L, T c,o T h,i 60C q C c (T c,o T c,i ) (kw /K)(60C 45C) 30 kw q q max 30kW 30kW 1 4

5 . (30 oint) You are meauring the combution ga temerature at the turbine inlet in a ga turbine engine uing a thermocoule junction. The thermocoule ha a diameter of 1 mm, and i initially at T i = 5 C. You may neglect radiation in the following analyi. Hot combution ga Thermocoule (evaluated at the free tream temerature) k = 0.04 W/mK k = 300 W/mK ρ = 0.6 kg/m 3 ρ = 9000 kg/m 3 c = 1000 J/kgK c = 400 J/kgK ν = 50 x 10-6 m / U = 0 m/ μ = 5 x 10-6 kg/m (ga vicoity evaluated at the thermocoule urface temerature, T ) (i) Determine the average convective heat tranfer coefficient, h. h = W/m K (ii) The temerature meaured at the thermocoule centerline after 3 ec i 900 C. Determine the combution ga temerature, T. T = K 5

6 Solution (i) Nu D (0.4 Re D 1/ 0.06Re D / 3 ) Pr 0.4 ( ) 1/ 4 Re D U L c U D U (0m /) (0.001m) m / Pr c k ( m /) (0.6 kg/m3 ) (1000J /kg K) 0.04 W /m K 0.75 ( m /) (0.6 kg/m 3 ) kg/m Nu D (0.4 Re D 1/ 0.06 Re D / 3 ) Pr 0.4 ( ) 1/ 4 ( / / 3 )( )( 3105 kg/m kg/m )1/ Nu D h D k h Nu D k D W /m K 0.001m W /m K 6

7 (ii) h L r 0.001m c o Bi Lc b 4ac kolid m ( W / m K) ( ) hlc Bi k 300 W / m K olid Lumed caacitance analyi i valid. T T t ex ( ) T T i t Vc 4 t V r A 4 r h A Vc ( r ) c 3 rc (9000 kg / m ) ( m) (400 J / kg K) t 1.0 ec h A h(4 r ) 3 h 3 ( W / m K) At t 3 ec, T 900 C T T 900 C T 3 ec ex ( ) T T 5 C T 1.0 ec i T C 7

8 3. (40 oint) Fluid enter a duct of triangular cro-ection with a ma flow rate of m. The cro-ection i an equilateral triangle of ide. The fluid i heated at the wall with a heat flux q ax +b (W/m ) into the domain, where x i meaured from the entrance to the duct. Furthermore, a chemical reaction in the volume of the fluid caue a contant volumetric heat generation rate of q W/m 3. The inlet bulk temerature of the fluid i T mi. You are given that the fluid ha a contant denity and a contant ecific heat C. dx Flow x Differential control volume (a) By conidering the differential control volume hown in the figure, write an energy balance to derive a ymbolic exreion for a a function of a, b,, x, m, q and the hyical dx roertie of the fluid. dt m dx = (b) Derive a ymbolic exreion for the variation of T m with x in term of a, b,, x, m, q, the inlet bulk temerature T mi and the hyical roertie of the fluid. T m (x) = (c) I the exreion you derived in art (b) for T m (x) (i) alway valid for thi roblem? (ii) only valid for thi roblem if the flow i fully develoed but the temerature i develoing? (iii) only valid for thi roblem if the flow and temerature are fully-develoed? 8

9 Circle only one anwer, and jutify your choice. Solution (a) q T m q T m +dx dx 3 Perimeter = 3 Area = 3 4 9

10 E E E E in out gen tore E mc T q" (3 ) dx mc T ( a x b) (3 ) dx in m m E out mc Tm dx mc ( Tm dx) dx 3 E gen q V q Adx q dx 4 E tore 0 3 E in E out E gen mc Tm ( a x b) (3 ) dx mc ( Tm dx) q dx 0 dx 4 3 q (3 ) ( a x b) dx mc dx dx 0 dx 4 mc (3 ) ( a x b) dx 3 4 q dx mc 4mc (3 ) ( a x b) 3 q 10

11 (b) dx mc 4mc (3 ) ( a x b) 3 q Tm Tmi, d T m 0 x (3 ) ( a x b) 3 q dx mc 4mc x (3 ) a x 3 q 3 a x 3 q Tm Tm, i ( b x) x ( b x) x mc 4mc mc 4mc 0 3 a x 3 q Tm Tm, i ( b x) x mc 4mc (c) It i ALWAYS VALID for the roblem. The exreion wa derived uing a imle energy balance. Therefore, no aumtion about fully develoed flow or heat tranfer are neceary. In fact, if q varie with x, no thermally fully develoed flow i oible. 11